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Theorem funco 5561
Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funco  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )

Proof of Theorem funco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmo 5539 . . . . 5  |-  ( Fun 
G  ->  E* z  x G z )
2 funmo 5539 . . . . . 6  |-  ( Fun 
F  ->  E* y 
z F y )
32alrimiv 1738 . . . . 5  |-  ( Fun 
F  ->  A. z E* y  z F
y )
4 moexexv 2313 . . . . 5  |-  ( ( E* z  x G z  /\  A. z E* y  z F
y )  ->  E* y E. z ( x G z  /\  z F y ) )
51, 3, 4syl2anr 476 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  E* y E. z ( x G z  /\  z F y ) )
65alrimiv 1738 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  A. x E* y E. z ( x G z  /\  z F y ) )
7 funopab 5556 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }  <->  A. x E* y E. z ( x G z  /\  z F y ) )
86, 7sylibr 212 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
9 df-co 4949 . . 3  |-  ( F  o.  G )  =  { <. x ,  y
>.  |  E. z
( x G z  /\  z F y ) }
109funeqi 5543 . 2  |-  ( Fun  ( F  o.  G
)  <->  Fun  { <. x ,  y >.  |  E. z ( x G z  /\  z F y ) } )
118, 10sylibr 212 1  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1401   E.wex 1631   E*wmo 2237   class class class wbr 4392   {copab 4449    o. ccom 4944   Fun wfun 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-fun 5525
This theorem is referenced by:  fnco  5624  f1co  5727  curry1  6828  curry2  6831  tposfun  6926  fsuppco  7813  fsuppco2  7814  fsuppcor  7815  fin23lem30  8672  smobeth  8911  hashkf  12359  xppreima  27811  comptiunov2i  35649  funresfunco  37545
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